Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-13T05:19:31.792Z Has data issue: false hasContentIssue false
  • English
  • Français

CALCULATION OF CRITICAL PARAMETERS FOR SPONTANEOUS COMBUSTION FOR SOME COMPLEX GEOMETRIES USING AN INDIRECT NUMERICAL METHOD

Part of: Partial differential equations Numerical problems in dynamical systems Partial differential equations, boundary value problems

Published online by Cambridge University Press:  26 February 2018

QUANBING LUO Open the ORCID record for QUANBING LUO [Opens in a new window] ,
DONG LIANG ,
TING REN  and
JIAN ZHANG
QUANBING LUO*
Affiliation:
School of Environmental Science and Engineering, Guangdong University of Technology, Guangzhou 510006, China email quanbing.luo@foxmail.com
DONG LIANG
Affiliation:
School of Engineering, Sun Yat-sen University, Guangzhou 510006, China email gzliangd@163.com Guangdong Provincial Key Laboratory of Fire Science and Technology, Guangzhou 510006, China
TING REN
Affiliation:
School of Civil, Mining and Environmental Engineering, University of Wollongong, Wollongong 2522, Australia email tren@uow.edu.au, jz164@uowmail.edu.au
JIAN ZHANG
Affiliation:
School of Civil, Mining and Environmental Engineering, University of Wollongong, Wollongong 2522, Australia email tren@uow.edu.au, jz164@uowmail.edu.au
*
✉quanbing.luo@foxmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the theory of spontaneous combustion, identifying the critical value of the Frank-Kamenetskii parameter corresponds to solving a bifurcation point problem. There are two different numerical methods used to solve this problem—the direct and indirect numerical methods. The latter finds the bifurcation point by solving a partial differential equation (PDE) problem. This is a better method to find the bifurcation point for complex geometries. This paper improves the indirect numerical method by combining the grid-domain extension method with the matrix equation computation method. We calculate the critical parameters of the Frank-Kamenetskii equation for some complex geometries using the indirect numerical method. Our results show that both the curve of the outer boundary and the height of the geometries have an effect on the values of the critical Frank-Kamenetskii parameter, however, they have little effect on the critical dimensionless temperature.

Keywords

critical parametersFrank-Kamenetskiiindirect numerical methodbifurcation pointspontaneous combustion

MSC classification

Primary: 65P30: Bifurcation problems
Secondary: 65N06: Finite difference methods 35B32: Bifurcation
Type
Research Article
Information
The ANZIAM Journal , Volume 59 , Issue 3 , January 2018 , pp. 402 - 412
Copyright
© 2018 Australian Mathematical Society 

References

Anderson, C. and Zienkiewicz, O., “Spontaneous ignition: finite element solutions for steady and transient conditions”, J. Heat Transfer 96 (1974) 398–404; doi:10.1115/1.3450212.CrossRefGoogle Scholar
Anderson, J. D., Computational fluid dynamics: the basics with applications (McGraw Hill, New York, 1995).Google Scholar
Bartels, R. H. and Stewart, G. W., “Solution of the matrix equation $ax+xb=c$ [f4]”, Commun. ACM 15 (1972) 820–826; doi:10.1145/361573.361582.Google Scholar
Boddington, T., Gray, P. and Harvey, D. I., “Thermal theory of spontaneous ignition: criticality in bodies of arbitrary shape”, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 270 (1971) 467–506; doi:10.1016/j.amc.2011.06.009.Google Scholar
Bowes, P. C., Self-heating: evaluating and controlling the hazards (Elsevier, Amsterdam, 1984).Google Scholar
Du, X. and Feng, C., “Critical parameters for the thermal explosion of exothermic systems having two-dimensional geometries (in Chinese)”, J. Beijing Inst. Tech. 14 (1994) 37–42.Google Scholar
Frank-Kamenetskii, D. A., Diffusion and heat transfer in chemical kinetics (Plenum Press, New York, 1969).Google Scholar
Golub, G., Nash, S. and Van Loan, C., “A Hessenberg–Schur method for the problem $ax+xb=c$ ”, IEEE Trans. Automatic Control 24 (1979) 909–913; doi:10.1109/TAC.1979.1102170.CrossRefGoogle Scholar
Gray, B., “Spontaneous combustion and self-heating”, in SFPE handbook of fire protection engineering, 5th edn (eds Hurley, M. J. et al. ), (Springer, New York, 2016) 604–632; doi:10.1007/978-1-4939-2565-0.Google Scholar
Luo, Q., Liang, D. and Mo, S., “Numerical calculation of the critical parameters of Frank-Kamenetskii equation in spontaneous combustion theory”, Numer. Heat Transfer, Part B: Fundamentals 68 (2015) 403–417; doi:10.1080/10407790.2015.1036625.Google Scholar
Moore, G. and Spence, A., “The calculation of turning points of nonlinear equations”, SIAM J. Numer. Anal. 17 (1980) 567–576; doi:10.1137/0717048.Google Scholar
Partridge, P. W. and Wrobel, L. C., “The dual reciprocity boundary element method for spontaneous ignition”, Int. J. Numer. Methods Eng. 30 (1990) 953–963; doi:10.1002/nme.1620300502.Google Scholar
Prata, A. T. and Sparrow, E. M., “Heat transfer and fluid flow characteristics for an annulus of periodically varying cross section”, Numer. Heat Transfer 7 (1984) 285–304; doi:10.1080/01495728408961826.CrossRefGoogle Scholar
Roose, D. and Hlavacek, V., “Numerical computation of Hopf bifurcation points for parabolic diffusion-reaction differential equations”, SIAM J. Appl. Math. 43 (1983) 1075–1085; doi:10.1137/0143070.CrossRefGoogle Scholar
Roose, D. and Hlavacek, V., “A direct method for the computation of Hopf bifurcation points”, SIAM J. Appl. Math. 45 (1985) 879–894; doi:10.1137/0145053.CrossRefGoogle Scholar
Roose, D., Piessens, R., Hlavacek, V. and van Rompay, P., “Direct evaluation of critical conditions for thermal explosion and catalytic reaction”, Combust. Flame 55 (1984) 323–329; doi:10.1016/0010-2180(84)90171-8.Google Scholar
Sexton, M. J., Macaskill, C. and Gray, B. F., “Thermal ignition in rectangular and triangular regions”, ANZIAM J. 42(E) (2000) C1283–C1304; doi:10.0000/anziamj.v42i0.646.Google Scholar
Seydel, R., “Numerical computation of branch points in nonlinear equations”, Numer. Math. 33 (1979) 339–352; doi:10.1007/BF01398649.Google Scholar
Sorensen, D. C. and Zhou, Y., “Direct methods for matrix sylvester and lyapunov equations”, J. Appl. Math. 2003 (2003) 277–303; doi:10.1155/S1110757X03212055.Google Scholar
Tao, W., Numerical heat transfer, 2nd edn (Xi’an Jiaotong University Press, 2001) (in Chinese).Google Scholar
Xue, D. and Chen, Y., Solving applied mathematical problems with MATLAB (Chapman and Hall/CRC, Boca Raton, 2008); doi:10.1201/b17177.Google Scholar
Zhang, Y. P., Su, G. H., Qiu, S. Z., Tian, W. X., Gaus-Liu, X., Kretzschmar, F. and Miassoedov, A., “Numerical study on the heat transfer characteristics of live-l4 melt pool with a partial solidification process”, Prog. Nucl. Energy 74 (2014) 213–221; doi:10.1016/j.pnucene.2014.03.011.Google Scholar